In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 6 cos θ + 4 sin θ

Verified step by step guidance

1. Start by recalling the relationships between polar and rectangular coordinates: $x = r \cos \theta$ and $y = r \sin \theta$. Also, $r^2 = x^2 + y^2$ and $\tan \theta = \frac{y}{x}$. These will help in converting the polar equation to a rectangular form.
2. Given the polar equation $r = 6 \cos \theta + 4 \sin \theta$, multiply both sides by $r$ to utilize the identity $r^2 = x^2 + y^2$. This gives $r^2 = r(6 \cos \theta + 4 \sin \theta)$.
3. Substitute $r^2 = x^2 + y^2$, $r \cos \theta = x$, and $r \sin \theta = y$ into the equation. This results in $x^2 + y^2 = 6x + 4y$.
4. Rearrange the equation to bring all terms to one side: $x^2 + y^2 - 6x - 4y = 0$. This is the rectangular form of the given polar equation.
5. To graph the equation, complete the square for both $x$ and $y$ terms. This will help identify the conic section represented by the equation.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to interpret and manipulate these coordinates is essential for converting polar equations to rectangular form.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes (x and y) to define the position of points in a plane. The conversion from polar to rectangular coordinates involves using the relationships x = r cos(θ) and y = r sin(θ). This understanding is crucial for graphing equations in the rectangular coordinate system.

Conversion Formulas

The conversion from polar to rectangular coordinates relies on specific formulas that relate the two systems. For a polar equation given by r and θ, the rectangular coordinates can be derived using x = r cos(θ) and y = r sin(θ). Mastery of these formulas allows for the effective transformation of polar equations into a format suitable for graphing in the rectangular coordinate system.